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Fourier‐transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier–Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.

Item Type:	Refereed Article
Keywords:	aperiodicity, pure point measures, Fourier transform
Research Division:	Mathematical Sciences
Research Group:	Pure mathematics
Research Field:	Lie groups, harmonic and Fourier analysis
Objective Division:	Expanding Knowledge
Objective Group:	Expanding knowledge
Objective Field:	Expanding knowledge in the mathematical sciences
UTAS Author:	Terauds, V (Dr Venta Terauds)
ID Code:	140823
Year Published:	2020
Deposited By:	Mathematics
Deposited On:	2020-09-10
Last Modified:	2020-10-09
Downloads:	9 View Download Statistics